Integrand size = 17, antiderivative size = 148 \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3}+\frac {5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac {10 b^2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^{3/2}}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}} \]
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Time = 0.06 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {272, 44, 59, 631, 210, 31} \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {10 b^2 \arctan \left (\frac {2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3} \]
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Rule 31
Rule 44
Rule 59
Rule 210
Rule 272
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^{2/3}} \, dx,x,x^{3/2}\right ) \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{2/3}} \, dx,x,x^{3/2}\right )}{9 a} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3}+\frac {5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}+\frac {\left (10 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^{3/2}\right )}{27 a^2} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3}+\frac {5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{7/3}} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3}+\frac {5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}}+\frac {\left (10 b^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^{3/2}}}{\sqrt [3]{a}}\right )}{9 a^{8/3}} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{3 a x^3}+\frac {5 b \sqrt [3]{a+b x^{3/2}}}{9 a^2 x^{3/2}}-\frac {10 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^{3/2}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{8/3}}-\frac {5 b^2 \log (x)}{18 a^{8/3}}+\frac {5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{9 a^{8/3}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\frac {3 a^{2/3} \sqrt [3]{a+b x^{3/2}} \left (-3 a+5 b x^{3/2}\right )}{x^3}-10 \sqrt {3} b^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^{3/2}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 b^2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^{3/2}}\right )-5 b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^{3/2}}+\left (a+b x^{3/2}\right )^{2/3}\right )}{27 a^{8/3}} \]
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\[\int \frac {1}{x^{4} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 3.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=- \frac {2 \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{\frac {3}{2}}}} \right )}}{3 b^{\frac {2}{3}} x^{4} \Gamma \left (\frac {11}{3}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {10 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{27 \, a^{\frac {8}{3}}} - \frac {5 \, b^{2} \log \left ({\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} + {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{27 \, a^{\frac {8}{3}}} + \frac {10 \, b^{2} \log \left ({\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{27 \, a^{\frac {8}{3}}} + \frac {5 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} b^{2} - 8 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a b^{2}}{9 \, {\left ({\left (b x^{\frac {3}{2}} + a\right )}^{2} a^{2} - 2 \, {\left (b x^{\frac {3}{2}} + a\right )} a^{3} + a^{4}\right )}} \]
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Time = 1.55 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {\frac {10 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {8}{3}}} + \frac {5 \, b^{3} \log \left ({\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} + {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {8}{3}}} - \frac {10 \, b^{3} \log \left ({\left | {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {8}{3}}} - \frac {3 \, {\left (5 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} b^{3} - 8 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a b^{3}\right )}}{a^{2} b^{2} x^{3}}}{27 \, b} \]
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Time = 6.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^4 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {10\,b^2\,\ln \left ({\left (a+b\,x^{3/2}\right )}^{1/3}-a^{1/3}\right )}{27\,a^{8/3}}-\frac {\frac {8\,b^2\,{\left (a+b\,x^{3/2}\right )}^{1/3}}{9\,a}-\frac {5\,b^2\,{\left (a+b\,x^{3/2}\right )}^{4/3}}{9\,a^2}}{{\left (a+b\,x^{3/2}\right )}^2-2\,a\,\left (a+b\,x^{3/2}\right )+a^2}+\frac {\ln \left (\frac {-5\,b^2+\sqrt {3}\,b^2\,5{}\mathrm {i}}{3\,a^{5/3}}-\frac {10\,b^2\,{\left (a+b\,x^{3/2}\right )}^{1/3}}{3\,a^2}\right )\,\left (-5\,b^2+\sqrt {3}\,b^2\,5{}\mathrm {i}\right )}{27\,a^{8/3}}-\frac {\ln \left (\frac {5\,b^2+\sqrt {3}\,b^2\,5{}\mathrm {i}}{3\,a^{5/3}}+\frac {10\,b^2\,{\left (a+b\,x^{3/2}\right )}^{1/3}}{3\,a^2}\right )\,\left (5\,b^2+\sqrt {3}\,b^2\,5{}\mathrm {i}\right )}{27\,a^{8/3}} \]
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